CFD | RANS vs. URANS vs. LES

Characteristics of turbulence

One characteristic of turbulent flows is their irregularity (or randomness). A fully deterministic approach to characterize turbulent flows is very difficult. Turbulent flows are usually described statistically.

• The diffusivity of turbulence cause rapid mixing and increased rates of momentum, heat, and mass transfer. A flow that looks random but does not exhibit the spreading of velocity fluctuations through the surrounding fluid is not turbulent.
• Turbulent flows are dissipative. Kinetic energy gets converted into heat due to viscous shear stresses. Turbulent flows die out quickly when no energy is supplied.
• Turbulent flows always occur at high Reynolds numbers. They are caused by a complex interaction between the viscous forces and convection.
• Turbulent flows are rotational, that is, they have non-zero vorticity. Mechanisms such as the stretching of three-dimensional vortices play a key role in turbulence.
• Turbulence is a continuum phenomenon. Even the smallest eddies are significantly larger than the molecular scales.
• Turbulence is a feature of flow and is not a property of the fluid. A liquid or a gas at high Reynolds number will exhibit the same dynamics.

1. Turbulent flows can originate at the walls. When this is the case, we talk about wall bounded turbulence.
2. Turbulent flows can also originate in the absence of walls (or far from walls). When this is the case, we talk about shear free turbulence (usually jets, heated walls, atmospheric flows).

Instantaneous fluctuations – Removing small scales

We have seen that turbulent flows are characterize by instantaneous fluctuations of velocity, pressure, and all transported quantities.

• In most engineering applications is not of interest resolving the instantaneous fluctuations.
• To avoid the need to resolve the instantaneous fluctuations (or small scales), two methods can be used:
  • Reynolds averaging.
  • Filtering.
• If you want to resolve all scales, you conduct DNS simulations, which are very computationally intensive.
• Two methods can be used to eliminate the need to resolve the small scales:
  • Reynolds averaging (RANS/URANS):
    • All turbulence scales are modeled.
    • Can be 2D and 3D.
    • Can be steady or unsteady.
  • Filtering (LES/DES):
    • Resolves large eddies.
    • Models small eddies.
    • Intrinsically 3D and unsteady.
• Both methods introduce additional terms in the governing equations that must be modeled (these terms are related to the instantaneous fluctuations).
• The final goal of turbulence modeling is to find the closure equations to model these additional terms (usually a stress tensor).

Reynolds Averaging (RANS)

Averaging the quantities (time average, spatial average or ensemble average)

• Reynolds averaging simple consists in:
  • Splitting the instantaneous value of the primitive variables into a mean component and a fluctuating component (Reynolds decomposition).
  • Averaging the quantities (time average, spatial average or ensemble average).
  • Applying a few averaging rules to simplify the equations
  • When we use Reynolds averaging, we are taking a statistical approach to turbulence modeling.

1. RANS (Reynolds-Averaged Navier-Stokes): Time averaging
2. URANS (Unsteady Reynolds-Averaged Navier-Stokes): Ensemble averaging

Time averaging

Time averaging is appropriate for stationary turbulence or slowly varying turbulent flows.

$$

Spatial averaging

Spatial averaging is appropriate for homogenous turbulence.

$$

Ensemble averaging

Ensemble averaging is appropriate for unsteady turbulence.

• In ensemble averaging, the number or realizations (or experiments) must be large enough to eliminate the effects of fluctuations. This type of averaging can be used with steady or unsteady flows.
• If you can afford it, ensemble averaging is recommended. Have in mind that CFD is deterministic, so you should start each realization using different initial conditions and boundary conditions fluctuations to obtain different outcomes.

$$

Averaging rules

Question: URANS

• URANS we basically insert the time derivative and ensemble-average the flow for a certain time step (see the picture, which is taken from the book of Ferzinger, 2020; left side is supposed to be RANS and right side is supposed to be URANS).
• URANS is a Reynolds-Averaged equation. When you do this reynolds-averaging you assume that you average long enough so that you resolve the mean. That is, you assume you CAN actually find the average. If the flow time-scales and turbulent time-scales are similar and you don't average long enough the turbulent contribution will not be zero and you therefore get a mean-flow that is contaminated by turbulence statistics (which is not the true mean flow). This to me in a user-error and not a problem with URANS. If you don't sample long enough to get the correct statistics then you simply get the wrong statistics.link
• I think the definition of URANS is contentious for CFD theorists and therefore so is its applicability. In general, URANS isn't really defined very well. I've seen some try to define it in a similar way to LES in terms of how the energy spectrum is captured/modelled. I think this discussion really becomes important (useful) when developing hybrid resolving/modelling method (DES). Moreover, most if not all the commercial codes do not explain the foundation of how their URANS approuch is implimented. I'm sure it's usually just ensuring there's a time derivative in the transport equations. But agree with other posts here, for all intents and purposes, transient RANS ~ URANS. link
• 对于 $$ 的 $$，从理论上 URANS 这个$$是任意的，如果不考虑附加的湍流封闭模型如ke之类，$$足够的小，就变成了LES (LES can be applied spatial/temporal/both filtering)，$$足够的大，就变成了RANS。然而$$ 并不需要显性的给定。所以对URANS取决定性的还是 $$. link
  • 类似的，URANS与LES从传输方程上区别也非常少，区别仅仅在与 $$ 的模化，如果是LES的 $$ ，那就是LES，如果是URANS的 $$ ，那就是URANS。湍流理论会从底层特别深的地方，推导出来 $$ 的不同，但是代码就是代码，不知道底层啥样的，到了代码这一层，只不过就是 $$ 的大小不 同。
• In below figure (right), let's define $$ for URANS.
• Question: What turbulence model does URANS use? 1) Same as RANS?

• link: 雷诺时均法对非稳态NS方程做平均以后，多出了反应湍流脉动的雷诺应力项，需要通过模型封闭。常用的封闭方法是基于Boussineq假设（又即，涡粘假设），该假设将雷诺应力与应变率的关系视为正比，其比例系数即为湍流粘性系数（又即，涡粘系数）。之后，通过对湍流粘性系数进行建模对NS方程进行封闭。某一变量的雷诺平均操作如下： $$, 其中时间间隔比较重要，物理上要求这个时间间隔要远大于湍流的随机脉动周期，但又要远小于时均量的变化周期 —— 这是一个令人直观上感觉很奇怪的要求。在时均值几乎不随时间变化的稳态或准稳态湍流中，时间间隔趋近于无穷大；但在时均值随时间变化的非定常湍流中，时间间隔的取值非常重要。如果时间间隔取值太小，则平均量无法代表该段时间内物理量的整体均值；而如果时间间隔取值过大，则求解结果无法反应流动的变化细节，这是制约URANS计算结果精度的原因之一。LES采用非稳态NS方程直接对大涡进行模拟，而采用亚格子模型来对小涡进行建模。亚格子模型建模的思路却与RANS中的Boussineq假设相似，即将湍流脉动所造成的影响用湍流粘性系数来描述，之后通过模型封闭湍流粘性系数进而封闭整个控制方程。
  • 所不同的是，LES方法只是对亚格子尺度上的湍流脉动进行建模，
  • 而RANS则是对整场尺度的湍流脉动进行建模，
  • 而这种整场的模化必然也会导致URANS计算结果精度存在一定偏差。
• link 雷诺平均，经常被误解为时间平均，其实应该是样本平均。只是在不同的条件下，样本平均可以用这样那样的平均方式代替。从RANS和URANS的平均方式与求解结果的验证方法上来看：
  • 1.若流动“时间统计”均匀，也即T趋于无穷大时，T上的时间平均存在，且时均值与时间无关。对应的是RANS，因此可以用实验数据做样本平均（或很长时间的时间平均）跟RANS结果对比。可以肯定的是：RANS计算出的结果是流场变量的系统平均。
  • 2.若流场存在缓变周期T2（imposed unsteadiness），湍流脉动周期为T1，且T1，T2满足这样的条件：存在T对于T2来说无穷小，对于T1来说无穷大，那么可以在T上作时间平均得到URANS。对于实验数据与仿真数据的对比，当比较均值时，可以用实验数据在T上作平均跟URANS结果对比；若要对比脉动值，则需要把URANS捕捉到的脉动（URANS结果减去URANS结果的时间平均值（该值应该与时间无关））跟湍流模型模拟的脉动值加起来跟实验脉动数据（实验值减去它的二次时间平均）对比。
  • 3.若不满足上面那条，但是流动属周期流动的，则可以通过相平均得到URANS，此时可用实验数据做相平均与URANS的结果对比，且可以通过判断URANS结果的时间平均是否与时间无关来确定此种流动是否合适用URANS。对于实验数据与仿真数据的对比，当比较均值时，可以用实验数据做相平均作跟URANS结果直接对比；若要对比脉动值，则需要把URANS捕捉到的脉动（URANS结果减去URANS结果的时间平均值（该值也应该与时间无关））跟湍流模型模拟的脉动加起来跟实验脉动数据（实验值减去其相平均值的时均值）对比。
• 若上面的都不满足（可以用URANS的结果做时间平均后是否随时间变化来确定），就只能老老实实的用系统平均，或者LES或DNS了。

Question: What is phase averaging?

Large eddy simulation (LES)

Large eddy simulation (LES) is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, and first explored by Deardorff (1970).

An LES filter can be applied to a spatial and temporal field $$ and perform a spatial filtering operation, a temporal filtering operation, or both. The filtered field, denoted with a bar, is defined as

$$

$$

$$

Smagorinsky–Lilly model

The first SGS model developed was the Smagorinsky–Lilly SGS model, which was developed by Joseph Smagorinsky and used in the first LES simulation by Deardorff. It models the eddy viscosity as:

$$

where $$ is the grid size and $$ is a constant.

This method assumes that the energy production and dissipation of the small scales are in equilibrium.

Ex: 1D Burgers equation

Consider the 1D Burgers equation in terms of the velocity field $$ $$

LES

In the continuous LES approach the scale separation is obtained via low-pass filtering. The spatial filtering operation (no time-filtering is considered here) can be expressed as the convolution product between the unfiltered point-wise velocity field and some suitable filter function $$, that is $$ $$ being the characteristic filter width. The LES equation is obtained by applying (2) to each term of (1) $$

Commutation between filtering and derivative being applied

URANS

As a counterpart, in the continuous URANS approach a statistical averaging is adopted. $$

$$ being the number of sample of the ensemble averaging. The URANS equation is obtained by applying (4) to each term of (1)

$$

Close equations

In order to close Eqs.(3) and (5), suitable models are used, that is

$$

Discretization

If we consider now the practical solution, a discretization of the derivative is introduced. Assume the same discretization is adopted for the LES and URANS equations, so that we can write a unique equation $$ that characterizes LES or URANS depending on the closure model.

One can deduce that the discrete equation (6) never implied a real application of a spatial filtering or an ensemble averaging. The only assessment is the spatial filtering implicitly induced by the discretization, a fact that is congruent to the LES assumption but not to the URANS one.

The Eq.(6) must be well posed by the set of initial and boundary conditions. This point adds a distinction as they should be different for LES and URANS.

References

1. Joel Guerrero's homepage
   1. 2tur_preliminaries (recommend)
   2. 3turbulence_Scales_law_of_the_wall_wall_function (recommend)
   3. 5governing_equations (recommend)
2. turbulence_2021_OF8 (recommend)
3. URANS vs. LES
4. 雷诺平均 Navier-Stokes 方程